Question

find the indicated term of each expansion.
seventh term of ((a + b)^{10})
(\bigcirc) (a^{4}b^{6})
(\bigcirc) (210a^{7}b^{7})
(\bigcirc) (210a^{3}b^{4})
(\bigcirc) (210a^{4}b^{6})

question 9
1 pts
find the indicated term of each expansion.
eighth term of ((x - y)^{11})
(\bigcirc) (-330x^{4}y^{7})
(\bigcirc) (330x^{4}y^{7})
(\bigcirc) (x^{3}y^{4})
(\bigcirc) (x^{8}y^{8})

Explanation:

First Sub - Question: Seventh term of \((a + b)^{10}\)

Step 1: Recall the binomial theorem formula

The binomial theorem states that the \((r + 1)\)-th term of \((a + b)^n\) is given by \(T_{r+1}=\binom{n}{r}a^{n - r}b^{r}\). For the seventh term, \(r+1 = 7\), so \(r=6\). Here, \(n = 10\), \(a=a\), \(b = b\).

Step 2: Calculate the binomial coefficient

The binomial coefficient \(\binom{n}{r}=\frac{n!}{r!(n - r)!}\). Substituting \(n = 10\) and \(r = 6\), we have \(\binom{10}{6}=\frac{10!}{6!(10 - 6)!}=\frac{10!}{6!4!}=\frac{10\times9\times8\times7}{4\times3\times2\times1}=210\).

Step 3: Calculate the powers of \(a\) and \(b\)

For \(a\), the power is \(n-r=10 - 6 = 4\), so \(a^{4}\). For \(b\), the power is \(r = 6\), so \(b^{6}\).

Step 4: Combine the coefficient and the powers

The seventh term is \(\binom{10}{6}a^{10 - 6}b^{6}=210a^{4}b^{6}\).

Second Sub - Question: Eighth term of \((x - y)^{11}\)

Step 1: Recall the binomial theorem formula for \((a + b)^n\) (here \(a=x\), \(b=-y\), \(n = 11\))

The \((r + 1)\)-th term of \((a + b)^n\) is \(T_{r+1}=\binom{n}{r}a^{n - r}b^{r}\). For the eighth term, \(r + 1=8\), so \(r = 7\).

Step 2: Calculate the binomial coefficient

\(\binom{n}{r}=\binom{11}{7}=\frac{11!}{7!(11 - 7)!}=\frac{11!}{7!4!}=\frac{11\times10\times9\times8}{4\times3\times2\times1}=330\).

Step 3: Calculate the powers of \(a\) and \(b\)

For \(a=x\), the power is \(n-r=11 - 7 = 4\), so \(x^{4}\). For \(b=-y\), the power is \(r = 7\), so \((-y)^{7}=-y^{7}\).

Step 4: Combine the coefficient and the powers

The eighth term is \(\binom{11}{7}x^{11 - 7}(-y)^{7}=330\times x^{4}\times(-y^{7})=- 330x^{4}y^{7}\).

Answer:

s:

• Seventh term of \((a + b)^{10}\): D. \(210a^{4}b^{6}\)
• Eighth term of \((x - y)^{11}\): A. \(-330x^{4}y^{7}\)